Can All Coefficients 'a' and 'b' Be Roots in Their Own Set of Quadratics?

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The discussion centers on the mathematical problem of determining whether all coefficients 'a' and 'b' can serve as roots in their respective sets of quadratics of the form x² + a_ix + b_i. The participants established that for n=2, it is feasible for all values of 'a' and 'b' to be roots. However, challenges arise when n>3 due to the absence of fixed positions for each coefficient, leading to complexities in ensuring that all values remain distinct while satisfying the quadratic equation.

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1. Homework Statement :
We have 'n' quadratics is the form x^2 +aix+bi
All values of A and b are different.
Is it possible to have all values of A and B as the roots of the n quadratics

2. The attempt at a solution:
Well. I know that all values of 'a' and 'b' must (if possible) be in one of the n (x-t)(x-s) where t and s are two values of a and/or b. I have proven that when n=2 its possible for all values, however i have difficulty with n>3
 
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Could you please clarify the question? Are there any other constraints to the question (like the roots must be real/complex)?

Why should there be a problem when n>3?
 
Well there are no other restraints. Only all the values of a and b are different. I assume that both complex and real numbers are possible.

The answer to your second question I was having difficulty with N>3 is because there are not fixed 'positions' for each value
 

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